At a very low-level intuitive idea, consider that a very short pulse in time would have a very large frequency span. Similarly, a very large antenna (many wavelengths long) is typically very directive; very short antennas are often more omnidirectional. So at first glance it makes a little bit of sense.

To understand the relationship, think about a simple aperture in an infinite metallic sheet. Let the fields within the aperture (the source fields), be Ea. This, in effect, are the near fields. How do you find the far fields from the aperture fields?

You can integrate across the aperture: suppose you want the fields at some large position R away from the aperture. This can be found from:
integral[ Ea (dot) e^(-j*k (dot) R )/R ]

The exponent term comes from the phase propagation, the 1/R is the power falloff of the fields due to distance. This integral is basically a Fourier transform - note that R is fixed. The fourier transform in this case maps the near field domain (the aperture locations) to the far field domain (the angles theta and phi in the far field).

Hence, its just math. If you work out the equations that take the near fields out to the far fields, it resembles a Fourier Transform if you factor it right.

At a very low-level intuitive idea, consider that a very short pulse in time would have a very large frequency span. Similarly, a very large antenna (many wavelengths long) is typically very directive; very short antennas are often more omnidirectional. So at first glance it makes a little bit of sense.

To understand the relationship, think about a simple aperture in an infinite metallic sheet. Let the fields within the aperture (the source fields), be Ea. This, in effect, are the near fields. How do you find the far fields from the aperture fields?

You can integrate across the aperture: suppose you want the fields at some large position R away from the aperture. This can be found from:
integral[ Ea (dot) e^(-j*k (dot) R )/R ]

The exponent term comes from the phase propagation, the 1/R is the power falloff of the fields due to distance. This integral is basically a Fourier transform - note that R is fixed. The fourier transform in this case maps the near field domain (the aperture locations) to the far field domain (the angles theta and phi in the far field).

Hence, its just math. If you work out the equations that take the near fields out to the far fields, it resembles a Fourier Transform if you factor it right.

Say, if I have an aperture field from a horn antenna and just want to "brute force" compute the far-field, doing as few approximations as possible, I would just do a double integral across the limits of the aperture multiplied with the complex exponential e^(-j*k (dot) R)/R ?

In this case, what would the "k" and "R" in the exponent look like? I guess I would have to do a coordinate transformation at some point along the process, since I have my horn aperture field in rectangular coordinates, would this be before or after the computation of the far-field?

Basically the math is the same whether you're integrating E or Js (surface current). You can see how the dot product gets factored in, and you basically end up with the Fourier Transform of the aperture distribution.

I've designed a standard pyramidal horn using the design procedure outlined in Balanis Antenna Theory - Analysis and Design, 3rd Ed. and used the dimensions from this design to calculate the aperture field of the horn and stored the field in a matrix by using the expression for the tangential E-field component on each location in the discretized aperture, i.e. the expression shown below:

Then, as per my understanding, it should be possible to propagate the aperture field to the far-field, viewed in spherical coordinates by implementing the following equations, where a1 and b1 are the dimensions of the horn:

The conversion to spherical coordinates is done using

and setting Ex and Ez to zero and the integrals are solved for every point defined by r, theta, phi. I simply let Matlab crunch these equations numerically.

The total E-field in the far-field should then just be given by farfield = sqrt(|Er|^2+|Etheta|^2+|Ephi|^2), right?

Am I getting it right here or not? The reason I ask is that when I compare with a simulation of the far-field for the exact same horn simulated in a EM-solver and monitoring only the E-field, I get a far-field result that differs from the Matlab result. So I'm trying to figure out where I am doing it wrong. And the first very helpful thing would be to know if I at least got the process right?

Thanks for your reply!

Hope to hear from you again!

Best Regards,

Last edited by kviksand81 on Fri Feb 27, 2015 10:33 am; edited 1 time in total